Method, apparatus and design procedure for controlling multi-input, multi-output (MIMO) parameter dependent systems using feedback LTI&#39; zation

ABSTRACT

A method and apparatus are provided for controlling a dynamic device having multi-inputs and operating in an environment having multiple operating parameters. A method of designing flight control laws using multi-input, multi-output feedback LTI&#39;zation is also provided. The method includes steps of: (i) determining coordinates for flight vehicle equations of motion; (ii) transforming the coordinates for the flight vehicle equations of motion into a multi-input linear time invariant system; (iii) establishing control laws yielding the transformed equations of motion LTI; (iv) adjusting the control laws to obtain a desired closed loop behavior for the controlled system; and (v) converting the transformed coordinates control laws to physical coordinates.

BACKGROUND OF THE INVENTION

[0001] I. Field of the Invention

[0002] The present invention relates to a method of designing controllaws (e.g., flight control laws in an airplane) by applying a techniquecalled Multi-Input, Multi-Output (MIMO) feedback LTI'zation, which isapplicable to solving a feedback control design problem for a class ofnonlinear and linear parameter dependent (“LPD”) dynamic systems, alsoknown as linear parameter varying (“LPV”), with multiple inputs andmultiple outputs. Feedback LTI'zation combines a co-ordinatestransformation and a feedback control law, the results of which cancelsystem parameter dependent terms and yield the transformed space openloop system linear time invariant (LTI). The present invention furtherrelates to using multi-input feedback LTI'zation to solve the controldesign problem associated with control systems for LPD dynamic devices.In particular, the invention is applied to a feedback control system forcontrolling a parameter dependent dynamic device (e.g., an airplane)with multiple control inputs.

[0003] II. Background

[0004] Design techniques used for solving feedback control designproblems can be divided into several classes. For example, two broadclasses are (1) Linear Time Invariant systems (herein after referred toas “LTI”) and (2) nonlinear systems. In the last four decades, LTIsystems have received a great deal of attention resulting in manywell-defined control design techniques. See, e.g., Maciejowski, J. M.,Multivariable Feedback Design, 1989, Addison-Wesley and Reid, J. G.,Linear System Fundamentals, 1983, McGraw-Hill, each incorporated hereinby reference. Nonlinear systems have, in contrast, received far lessattention. Consequently, a smaller set of techniques has been developedfor use in feedback control system design for nonlinear systems orlinear parameter dependent systems. As a result, control law design fornonlinear systems can be an arduous task. Typically, control lawsconsist of a plurality of equations used to control a dynamic device ina desirable and predictable manner. Previously, designing control lawsfor LPD systems using quasi-static LTI design techniques could requirean enormous amount of effort, often entailing weeks, if not months, oftime to complete a single full envelope design. For example, whendesigning a flight control law, designers must predict and then designthe control law to accommodate a multitude (often thousands) ofoperating points within the flight envelope (i.e., the operating orperformance limits for an aircraft).

[0005] Feedback Linearization (reference may be had to Isidori, A.,Nonlinear Control Systems, 2nd Edition, 1989, Springer-Verlag, hereinincorporated by reference), is applicable to control design for a broadclass of nonlinear systems, but does not explicitly accommodate systemparameter changes at arbitrary rates. Feedback LTI'zation, a techniqueused for rendering a control system model linear time invariant, forsingle input systems is outlined in the Ph.D. thesis of the inventor,Dr. David W. Vos, “Non-linear Control Of An Autonomous Unicycle Robot;Practical Issues,” Massachusetts Institute of Technology, 1992,incorporated herein by reference. This thesis extends FeedbackLinearization to explicitly accommodate fast parameter variations.However, the Ph.D. thesis does not give generally applicable solutionsor algorithms for applying feedback LTI'zation to either single input ormulti-input parameter dependent dynamic systems. U.S. Pat. No. 5,615,119(herein incorporated by reference, and hereafter the “′119”patent)addressed this problem, albeit in the context of failure detectionfilter design. In particular, the ′119 patent describes a fault tolerantcontrol system including (i) a coordinate transforming diffeomorphismand (ii) a feedback control law, which produces a control system modelthat is linear time invariant (a feedback control law which renders acontrol system model linear time invariant is hereinafter termed “afeedback LTI'ing control law”).

[0006] The ′119 patent encompasses fault detection and isolation andcontrol law reconfiguration by transforming various actuator and sensorsignals into a linear time invariant coordinate system within which anLTI failure detection filter can be executed, to thus provide acapability for failure detection and isolation for dynamic systems whoseparameters vary over time. That is, a detection filter may beimplemented in a so-called Z-space in which the system may berepresented as linear time invariant and is independent of the dynamicsystem parameters.

[0007] What is needed, however, is the further extension of the feedbackLTIzation control law principals in the ′119 patent to multi-inputparameter dependent systems. Furthermore, control system designers havelong experienced a need for a fast and efficient method of designingcontrol laws relating to parameter dependent nonlinear systems. Anefficient method of control law design is therefore needed. Similarly,there is also a need for a control system aimed at controlling such adynamic device with multiple control inputs.

SUMMARY OF THE INVENTION

[0008] The present invention specifically solves a Multi-Input feedbackLTI'zation problem, and shows a method for feedback control law designfor a parameter dependent dynamic device (e.g., an airplane) class ofsystems. Additionally, the present invention provides a control systemfor controlling a parameter dependent dynamic device with multipleinputs. The present invention is also applicable to the methods andsystems discussed in the above-mentioned ′119 patent (i.e., for failuredetection system design in the multi-input case). As a result of theconcepts of this invention, control system designers may now shave weeksor months off of their design time.

[0009] According to one aspect of the invention, an automatic controlsystem for controlling a dynamic device is provided. The device includessensors and control laws stored in a memory. The control system includesa receiving means for receiving status signals (measuring the statevector) and current external condition signals (measuring parametervalues) from the sensors, and for receiving reference signals. Alsoincluded is processing structure for: (i) selecting and applying gainschedules to update the control laws, wherein the gain schedulescorrespond to the current external conditions signals (parameter values)and are generated in a multi-input linear time invariant coordinatessystem; (ii) determining parameter rates of change and applying theparameter rates of change to update the control laws; (iii) applyingdevice status signal feedback to update the control laws; and (iv)controlling the device based on the updated control laws.

[0010] According to another aspect of the invention, a method fordesigning flight control laws using multi-input parameter dependentfeedback is provided. The method includes the following steps: (i)determining a coordinates system for flight vehicle equations of motion;(ii) transforming the coordinates system for the flight vehicleequations of motion into a multi-input linear time invariant system;(iii) establishing control criteria yielding the transformed coordinatesequations of motion LTI; (iv) adjusting the control criteria to obtain adesired closed loop behavior for the controlled system; and (v)converting the transformed coordinates control laws to physicalcoordinates.

[0011] According to still another aspect of the invention, a method ofcontrolling a dynamic device is provided. The device includingactuators, sensors and control laws stored in a memory. The methodincludes the following steps: (i) transforming device characteristicsinto a multi-input linear time invariant system; (ii) selecting andapplying physical gain schedules to the control laws, the gain schedulescorresponding to the current external condition signals; (iii)determining and applying parameter rates of change to update the controllaws; (iv) applying device status signal feedback to update the controllaws; (v) converting the transformed coordinates control laws tophysical coordinates; and (vi) controlling the device based on theupdated control laws.

[0012] Specific computer executable software stored on a computer orprocessor readable medium is also another aspect of the presentinvention. This software code for developing control laws for dynamicdevices includes: (i) code to transform device characteristics into amulti-input linear time invariant system; (ii) code to establish controlcriteria yielding the transformed coordinates equations of motion LTI;(iii) code to define a design point in the multi-input linear timeinvariant system; (iv) code to adjust the transformations to correspondwith the design point; and (v) code to develop a physical coordinatescontrol law corresponding to the adjusted transformations; and (vi) codeto apply reverse transformations to cover the full design envelope.

[0013] In yet another aspect of the present invention, a multi-inputparameter dependent control system for controlling an aircraft isprovided. The system includes receiving means for receiving aircraftstatus signals and for receiving current external condition signals. Amemory having at least one region for storing computer executable codeis also included. A processor for executing the program code isprovided, wherein the program code includes code to: (i) transform theaircraft characteristics into a multi-input linear time invariantsystem; (ii) select and apply gain schedules to flight control laws, thegain schedules corresponding to the current external condition signals;(iii) determine parameter rates of change, and to apply the parameterrates of change to the flight control laws; (iv) apply feedback from theaircraft status signals to the flight control laws; (v) convert thetransformed coordinates control laws to physical coordinates; and (vi)control the aircraft based on the updated flight control laws.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The present invention will be more readily understood from adetailed description of the preferred embodiments taken in conjunctionwith the following figures.

[0015]FIG. 1 is a perspective view of an aircraft incorporating anautomatic control system of the present invention.

[0016]FIG. 2 is a functional block diagram describing an algorithmaccording to the present invention.

[0017]FIG. 3 is a flowchart showing a software flow carried out in theflight control computer of FIG. 1.

[0018]FIG. 4 is a block diagram of the flight control computer, sensors,and actuators, according to the FIG. 1 embodiments.

[0019]FIGS. 5a and 5 b are overlaid discrete time step response plotsaccording to the present invention.

[0020]FIGS. 5c and 5 d are overlaid Bode magnitude plots according tothe present invention.

[0021]FIG. 6 is an S-plane root loci plot for the closed and open looplateral dynamics according to the present invention.

[0022] FIGS. 7-18 are 3-D plots of auto pilot gains vs. air density anddynamic pressure according to the present invention.

[0023] FIGS. 19-30 are numerical gain lookup tables according to thepresent invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0024] Described herein is a technique called Multi-Input Feedback(“FBK”) LTI'zation, which is applicable to solving feedback controldesign problems for a class of nonlinear and linear parameter dependentsystems with multiple inputs such as actuator commands. This techniqueaccommodates arbitrary changes and rates of change of system parameters,such as air density and dynamic pressure. As will be appreciated by oneof ordinary skill in the art, a subset of nonlinear systems, namelylinear parameter dependent (herein after referred to as “LPD”) systems,is one method of modeling real world dynamic systems. Control design forsuch systems is traditionally achieved using LTI (linear time invariant)design techniques at a number of fixed parameter values (operatingconditions), where at each operating condition the system's equations ofmotion become LTI. Gain scheduling by curve fitting between these designpoints, is then used to vary the gains as the operating conditions vary.Feedback LTI'zation gives a simple and fast method for full envelopecontrol design, covering any parameter value. In addition, the resultinggain schedules (as discussed below) are an automatic product of thisdesign process, and the closed loop system can be shown to be stable forthe full parameter envelope and for arbitrary rates of variation of thesystem parameters throughout the operating envelope, using these gainschedules and the feedback LTI'ing control law.

[0025] The process of applying Feedback LTI'zation to facilitatedesigning control laws involves several steps, including: (i)transforming the coordinates of equations of motion of a dynamic device(e.g., an airplane) into a so-called z-space; (ii) defining a controllaw which yields the transformed coordinates equations of motion lineartime invariant (LTI); and (iii) applying LTI design techniques to thetransformed coordinates mathematical model to yield a desired closedloop behavior for the controlled system, all as discussed below. Thethird step is achieved by (a) designing the feedback gains in physicalcoordinates at a selected operating condition; (b) using the coordinatestransformations to map these gains into z-space; and (c) reverse mappingvia the coordinates transformations and Feedback LTI'ing control laws todetermine physical coordinates control laws for operating conditionsother than at the design conditions.

[0026] One aspect of the present invention will be described withrespect to an aircraft automatic flight control system for maintainingdesired handling qualities and dynamic performance of the aircraft.However, the present invention is also applicable to other dynamicdevices, such as vehicles including automobiles, trains, and robots; andto other dynamic systems requiring monitoring and control. Furthermore,the present invention encompasses a design method and system fordesigning control laws by, for example, defining a point in z-space, andthen updating a system transformation to generate control laws inx-space (i.e., in physical coordinates).

[0027]FIG. 1 is a perspective view of an aircraft 1 having flightcontrol surfaces such as ailerons 101, elevator 102, and rudder 103. Forexample, aircraft 1 is a Perseus 004 unmanned aircraft operated byAurora Flight Science, Inc. of Manassas, Va. Each flight control surfacehas an actuator (not shown in FIG. 1) for controlling the correspondingsurface to achieve controlled flight. Of course, other flight controlactuators may be provided such as throttle, propeller pitch, fuelmixture, trim, brake, cowl flap, etc.

[0028] The actuators described above are controlled by a flight controlcomputer 104 which outputs actuator control signals in accordance withone or more flight control algorithms (hereinafter termed “flightcontrol laws”) in order to achieve controlled flight. As expected, theflight control computer 104 has at least one processor for executing theflight control laws, and/or for processing control software oralgorithms for controlling flight. Also, the flight computer may have astorage device, such as Read-Only Memory (“ROM”), Random Access Memory(“RAM”), and/or other electronic memory circuits.

[0029] The flight control computer 104 receives as inputs sensor statussignals from the sensors disposed in sensor rack 105. Various aircraftperformance sensors disposed about the aircraft monitor and providesignals to the sensor rack 105, which in turn, provides the sensorsignals to the flight control computer 104. For example, providedaircraft sensors may include: an altimeter; an airspeed probe; avertical gyro for measuring roll and pitch attitudes; rate gyros formeasuring roll, pitch, and yaw angular rates; a magnetometer fordirectional information; alpha-beta air probes for measuring angle ofattack and sideslip angle; etc. As will be appreciated, if a roll ratesensor is not included in the sensor suite or rack 105, a roll ratesignal may be synthesized by using the same strategy as would be used ifan onboard roll rate sensor failed in flight. Meaning that the roll ratesignal is synthesized by taking the discrete derivative of the rollattitude (bank angle) signal. The manipulation (i.e., taking thediscrete derivative) of the bank angle signal may be carried out bysoftware running on the flight computer 104. Thus, using sensor statusinputs, control algorithms, and RAM look-up tables, the flight controlcomputer 104 generates actuator output commands to control the variousflight control surfaces to maintain stable flight.

[0030]FIG. 2 is a functional block diagram illustrating functionalaspects according to the present invention. FIG. 2 illustrates analgorithm according to the present invention including flight controllaws for a two-control input (e.g., rudder and aileron), as well as atwo-parameter (e.g., air density and dynamic pressure) system. Inparticular, this algorithm controls the lateral dynamics of an unmannedaircraft (e.g., the Perseus 004 aircraft). As will be appreciated by oneof ordinary skill in the art, a system using more than two inputs ormore than two parameters is a simple extension of the principles andequations discussed below. Therefore, the invention is not limited to acase or system using only two inputs or two parameters. Instead, thepresent invention may accommodate any multiple input and multipleparameter system.

[0031] As is known, a mathematical model of the aircraft (e.g., aparameter-dependent dynamic system) depicted in FIG. 1 may be written ina physical coordinate system (hereinafter called a coordinate system inx-space). In the case of an aircraft, a Cartesian axis system may haveone axis disposed along the fuselage toward the nose, one axis disposedalong the wing toward the right wing tip, and one axis disposed straightdown from the center of mass, perpendicular to the plane incorporatingthe first two axes. Measurements via sensors placed along or about theseaxes provide information regarding, for example, roll rate, bank angle,side slip, yaw rate, angle of attack, pitch rate, pitch attitude,airspeed, etc.

[0032] The ′119 patent provides a description of the methodology forsolving the problem of finding a state space transformation and feedbackcontrol law for a single input linear parameter dependent system. Thesolutions to these problems yield linear parameter dependent (“LPD”;also referred to as linear parameter varying-“LPV”) coordinatestransformations, which, when applied to system model equations of motiontogether with Feedback LTI'ing control laws, yield descriptions that arelinear time invariant (LTI) in the transformed state space (z-space). Asdiscussed in the ′119 patent, a detection filter may be implemented inz-space in which the dynamic system (e.g., an aircraft) may berepresented as linear time invariant and, as such, is independent of thedynamic system parameters. Basically, “non-stationary” aircraft flightdynamics equations are transformed into “stationary” linear equations ina general and systematic fashion. In this context, “stationary” impliesthat the dynamic characteristics are not changing. As a result, a set ofconstant coefficient differential equations is generated in z-space formodeling the system. The combination of state space transformation andfeedback control law (this control law is called a “Feedback LTI'ingcontrol law”), which cancels all the parameter dependent terms, isreferred to as “Feedback LTI'zation”. By way of example, the solution asdescribed in U.S. Pat. No. ′119 would be applicable to control of thelongitudinal aircraft dynamics problem, using a single input (e.g., theelevator).

[0033] The present invention details the further extension of FeedbackLTI'zation to accommodate a multi-input case which is particularlyrelevant to the lateral axis of a conventional aircraft, using a rudderand ailerons as actuators, for example. As such, the present inventionencompasses a control system for controlling dynamic devices and afailure detection system. This formulation is also relevant to highperformance aircraft, which may have multiple actuators in both thelateral and longitudinal axes.

[0034] In physical terms, the present problem is similar to the onefaced in a single input case; namely, a mathematical description of thevehicle dynamics must be found, which does not change as the parametersof the vehicle change. In other words, it is desirable to rewrite asystem equations of motion such that the dynamic behavior of the systemis always the same and thus very predictable, regardless of what theoperating conditions or operating parameters are (e.g., the systemdynamics need to be expressed as linear time invariant (LTI) for anyparameter value or any rate of change of the parameter values).

[0035] This process of describing the equations of motion according tothe above requirements can be achieved through a combination ofcoordinate change and feedback control laws. A coordinatetransformation, or diffeomorphism, is determined which transforms thephysical coordinates (i.e., x-space) description of the aircraft dynamicmathematical model to a new set of coordinates (i.e., z-space). AFeedback LTI'ing control law is defined to cancel all the z-spaceparameter dependent terms, so that with this control law, the z-spacesystem is then independent of the parameters. In fact, the behavior ofthe model in the transformed coordinates is then that of a set ofintegrators, independent of vehicle operating conditions. It is thenpossible to prescribe the desired closed loop behavior by means of LTIcontrol design techniques, applied in x-space and transformed toz-space, which is then also valid for any point in the operatingenvelope (e.g., sea level to 22 km above sea level, and 20 m/s to 46.95m/s indicated airspeed (IAS)), and for arbitrary values and rates ofchange of the parameters defining the operating envelope. In this manneronly a single, or at worst, a small number of points in the operatingenvelope may be identified as design points. The diffeomorphism(transformation) has very specific dependence on the parameter values.By evaluating the parameter values and then evaluating thediffeomorphisms at these parameter values, the applicable gains areautomatically defined appropriate for the current operating conditions,and thus flight control laws may be obtained for use in the physicalcoordinates (x-space).

[0036] As will be appreciated by those skilled in the art, the processof designing control systems is fundamentally based on meetingrequirements for achieving desired closed loop characteristics orbehavior. Persons skilled in the art of LTI control design use wellknown techniques and follow standard procedures to achieve this. In themultivariable control domain for LTI systems, it is indeed possible toprescribe the desired closed loop dynamics of all the characteristicmotions of the vehicle, and a known algorithm such as “pole-placement”can be used to determine the gains that will deliver this. Other knowntechniques such as LQR (Linear Quadratic Regulator theory) can be usedto achieve the same goal.

[0037] As stated above, the coordinate change is mathematical, andallows a simple and easy mathematical treatment of the full envelopecontrol design problem. The Feedback LTI'ing control law can also bedescribed mathematically, however, it is implemented physically andinvolves a specific set of control algorithms. These control algorithms,combined with the coordinates change, result in the closed loopcontrolled physical dynamic behavior that is repeatable and predictableat all regions of the flight envelope.

[0038] The problem of finding such a coordinates change (diffeomorphism)and control law, is a central focus of both the feedback linearizationand feedback LTI'zation (for linear time varying parameters) problems.Essentially, the problem primarily requires solution of the coordinatestransformation matrix. After a solution is found, the rest of the designprocess follows as a matter of course. Clearly, this coordinate changeshould not result in any loss of information about the vehicle dynamicbehavior. In other words, the specific outputs combinations which ensurethat all the dynamic information is retained when observing the systembehavior from a different set of coordinates must be found. Finding thespecific outputs combinations has a direct relationship with transferfunctions, which describe how inputs reach outputs in a dynamic sense.That is, the solution is achieved when the new coordinates result in thefeature that all inputs will reach the outputs in a dynamic sense, andare not masked by internal system behavior. As will be appreciated, thecoordinates transformation should occur smoothly (e.g., without loss ofdata or without singularity) in both directions, i.e., from the model(z-space) description in one set of coordinates to the physicalcoordinates (x-space) description.

[0039] Once these output functions, or measurement directions, areknown, it becomes a fairly mechanical process to determine thecoordinates transformation and feedback control law, respectively. Infact, the present invention establishes that for LPD systems, the entireprocess of both finding the measurement direction, as well as thediffeomorphism and control laws, all become straightforward procedures,which can also be automated.

Solution of the Feedback LTI'zation Problem for LPD Multi-Input DynamicSystems

[0040] In order to define the diffeomorphism (Φ) and the FeedbackLTI'ing control law (ν), consider the affine multi-input parameterdependent system given by the equations: $\begin{matrix}{{\overset{.}{x} = {{f\left( {x,p} \right)} + {\sum\limits_{i = 1}^{m}{{g_{i}\left( {x,p} \right)}u_{i}}}}}\begin{matrix}{y_{1} = {h_{l}(x)}} \\\vdots \\{y_{m} = {h_{m}(x)}}\end{matrix}} & (1)\end{matrix}$

[0041] where xεR^(n); u_(i)yεR^(m); pεR^(q) and where x is a statevector made up of state variables, such as roll rate, yaw rate, sideslip, bank angle, etc., u_(i) is the i^(th) control input (e.g., rudderor aileron, etc.), and the output measurement directions, h_(i)(x), areto be determined appropriately in order to define the state variabletransformation. The functions f(.) and g(.) are functions of both thestate vector x as well as the parameter vector p.

[0042] First, a multivariable definition of relative degree is needed,namely a vector relative degree, which pertains to the number of zerosin the transfer function from the input vector u to the output vector y.This definition is taken directly from Isidori, discussed above.

[0043] Definition

[0044] A multivariable system of the form (1) with m inputs and moutputs, has vector relative degree {r₁, r₂, . . . , r_(m)} at a pointx_(o) if the following hold:

[0045] 1) For any 1≦i,j≦m and for all k≦r_(i)−1,

L _(gi) L _(f) ^(k) h _(i)(x)=0  (2)

[0046] where the operator “L” is the Lie derivative.

[0047] 2) The m by m matrix A(x) is nonsingular at x_(o), where:$\begin{matrix}{{A(x)} = \begin{bmatrix}{L_{g_{l}}L_{f}^{r_{i} - 1}{h_{l}(x)}} & \cdots & {L_{g_{m}}L_{f}^{r_{i} - 1}{h_{l}(x)}} \\\vdots & ⋰ & \vdots \\{L_{g_{l}}L_{f}^{r_{m} - 1}{h_{m}(x)}} & \cdots & {L_{g_{m}}L_{f}^{r_{m} - 1}{h_{m}(x)}}\end{bmatrix}} & (3)\end{matrix}$

[0048] The vector relative degree implies the multivariable notion ofthe system having no transfer function zeros, i.e., to ensure that nosystem characteristic dynamics information is lost by observing thesystem along the output directions col{h_(i)(x);h₂(x); . . . ;h_(m)(x)}.

[0049] State Space Exact Linearization for Multi input Systems

[0050] The state space exact linearization problem for multi-inputsystems can be solved if an only if there exists a neighborhood U ofx_(o) and m real valued functions h₁(x), h₂(x), . . . , h_(m)(x) definedon U, such that the system (1) has vector relative degree {r₁, r₂, . . ., r_(m)} at x_(o) and ${{\sum\limits_{i = 1}^{m}r_{i}} = n},$

[0051] with g(x_(o))=[g₁(x_(o)) g₂(x_(o)) . . . g_(m)(x_(o))] of rank m.

[0052] It remains to find the m output functions, h_(i)(x), satisfyingthese conditions in order to determine the state variabletransformation, given by the vectors:

φ_(k) ^(i)(x)=L _(f) ^(k−1) h _(i)(x) ∀1≦k≦r _(i), 1≦i≦m  (4)

[0053] then the diffeomorphism is constructed as: $\begin{matrix}{\Phi = \begin{bmatrix}{\varphi_{1}^{l}(x)} \\\vdots \\{\varphi_{r_{i}}^{l}(x)} \\\vdots \\{\varphi_{1}^{m}(x)} \\\vdots \\{\varphi_{r_{m}}^{m}(x)}\end{bmatrix}} & (5)\end{matrix}$

[0054] Solving for the m Output Functions for LPD Systems. Two InputsExample

[0055] It is directly demonstrated in this section how to find theoutput functions for the LPD lateral dynamics of the aircraft model,with rudder and aileron inputs, and all state variables (sideslip, rollrate, yaw rate and bank angle) measured. Of course, other inputs andstate variables could be solved for as well. In this case, m=2 and themodel can be written:

{dot over (x)}=Ax+Bu  (6)

[0056] with xεR⁴ and uεR². As will be appreciated by those of ordinaryskill in the art, variable A can represent an air vehicle dynamicsmatrix and variable B can represent a control distribution matrix.Variable u represents a vector of control, having variablescorresponding to the rudder and aileron, and x represents a system statevector (e.g., x=[side slip, bank angle, roll rate, and yaw rate]). Thevector relative degree of the system is {r₁, r₂}={2,2} and the summation${\sum\limits_{i = 1}^{m}r_{i}} = n$

[0057] is satisfied for r_(i)=2, m=2, and n=4. Evaluating the termsaccording to equation (2), yields for outputs y₁=C₁x and Y₂=C₂x:

C₁B₁=0

C₁B₂=0

C₂B₂=0

C₂B₂=0

C₁AB₁=1

C₁AB₂=0

C₂AB₁=0

C₂AB₂=1  (7)

[0058] where C_(i) is the i^(th) measurement direction. Note that thelower four equations of equation (7) satisfy the requirement thatequation (3) be nonsingular. These can be rearranged in matrix form:$\begin{matrix}{\begin{bmatrix}\left. \leftarrow\left. C_{1}\rightarrow \right. \right. \\\left. \leftarrow\left. C_{2}\rightarrow \right. \right.\end{bmatrix}\left\lbrack \begin{matrix}\underset{\downarrow}{\overset{\uparrow}{B_{1}}} & \underset{\downarrow}{\overset{\uparrow}{B_{2}}} & {\underset{\downarrow}{\overset{\uparrow}{A}}\quad \underset{\downarrow}{\overset{\uparrow}{B_{1}}}} & {\left. {A\quad B_{2}} \right\rbrack = \begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\end{matrix} \right.} & (8)\end{matrix}$

[0059] which then allows solution for C₁ and C₂, following which, thetransformation matrix, or diffeomorphism, can be written according toequations (4) and (5), as $\begin{matrix}{\Phi = \begin{bmatrix}\left. \leftarrow\left. C_{1}\rightarrow \right. \right. \\\left. \leftarrow\left. C_{2}\rightarrow \right. \right. \\\left. \leftarrow\left. {C_{1}A}\rightarrow \right. \right. \\\left. \leftarrow\left. {C_{2}A}\rightarrow \right. \right.\end{bmatrix}} & (9)\end{matrix}$

[0060] Feedback LTI'ing Control Law for a Two Input Lateral AircraftDynamics Model

[0061] The transformed z-space model is then determined, where z is thez-space state vector, Φ is the diffeomorphism, and x is the x-spacestate vector, from: $\begin{matrix}{z = {\Phi \quad x}} \\{\therefore} \\{\overset{.}{z} = {{\Phi \quad x} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}} \\{{= {{\Phi \quad A\quad x} + {\Phi \quad B\quad u} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}}} \\{{= {{\Phi \quad A\quad \Phi^{- 1}z} + {\Phi \quad B\quad u} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}}} \\{{= {{A_{z}z} + {B_{z}\quad u} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}}}\end{matrix}\quad$

[0062] which has the matrix form $\begin{matrix}{\begin{matrix}{\overset{.}{z} = {{\begin{bmatrix}{0\quad 0\quad 1\quad 0} \\{0\quad 0\quad 0\quad 1} \\\left. \leftarrow\left. {\alpha_{1}(z)}\rightarrow \right. \right. \\\left. \leftarrow\left. {\alpha_{2}(z)}\rightarrow \right. \right.\end{bmatrix}z} + {\begin{bmatrix}00 \\00 \\10 \\01\end{bmatrix}u} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}} \\{= {{\begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}z} + {\begin{bmatrix}00 \\00 \\10 \\01\end{bmatrix}v}}}\end{matrix}\quad} & (10)\end{matrix}$

[0063] where the new dynamics and control distribution matrices aredenoted A_(z) and B_(z), respectively. A new input is defined as followsto cancel the parameter dependent terms α₁(z) and α₂(z) and thesummation term which includes parameter rate of change terms (i.e.,representing the dependence of the coordinates transformation on therate at which the parameters change), thus yielding the feedback LTI'ingcontrol law, which together with the diffeomorphism of equation (9) hastransformed the LPD system (6) into an LTI system given by the last lineof (10). $\begin{matrix}{v = {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}\left\{ {{\begin{bmatrix}{0\quad 0\quad 0\quad 0} \\{0\quad 0\quad 0\quad 0} \\\left. \leftarrow\left. {\alpha_{1}(z)}\rightarrow \right. \right. \\\left. \leftarrow\left. {\alpha_{2}(z)}\rightarrow \right. \right.\end{bmatrix}z} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}} + {B_{z}u}} \right\}}} & (11)\end{matrix}$

[0064] This system (10) is now ready for application of LTI feedbackcontrol design, while the LTI'ing control law of (11) ensures that thesystem parameter dependence is accommodated in the physical control lawimplementation.

[0065] Solving for Gain Lookup Tables

[0066] The feedback LTI'ing control law (Eq. 11) for the two controlinput case is written as: $\begin{matrix}{v = {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}\left\{ {{\begin{bmatrix}0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\\leftarrow & \alpha_{1} & (z) & \rightarrow \\\leftarrow & \alpha_{2} & (z) & \rightarrow\end{bmatrix}z} + {\sum\limits_{i = 1}^{q}\quad {\frac{\partial\Phi_{x}}{\partial p_{i}}{\overset{.}{p}}_{i}}} + {B_{z}u}} \right\}}} & (12)\end{matrix}$

[0067] For the full state feedback z-space control law of the formν=−K_(z) z and ignoring the parameter rate of change terms forevaluating the lookup table gains (locally linear gains), therelationship between z-space locally linear gains and x-space locallylinear gains is as follows (where equivalently u=−K_(x)x):$\begin{matrix}\begin{matrix}{K_{x} = {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}\left\{ {\begin{bmatrix}0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\\leftarrow & \alpha_{1} & (z) & \rightarrow \\\leftarrow & \alpha_{2} & (z) & \rightarrow\end{bmatrix} + {B_{z}K_{z}}} \right\} \Phi}} \\{= {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}\left\{ {A_{z} + {B_{z}K_{z}}} \right\} \Phi}}\end{matrix} & (13)\end{matrix}$

[0068] From this expression, it will be obvious to one of ordinary skillin the art that the lookup gains may be stored as either x-space lookuptables (K_(x)), or as z-space lookup tables (K_(z)). In the case ofstoring the z-space gains, it is also necessary to store the z-spacematrices A_(z) and B_(z), as well as the diffeomorphism. This lattercase amounts to real time evaluation of z-space gains and thenconverting these to true physical space gains as opposed to performingthis transformation off-line and simply storing the x-space gains inlookup tables.

[0069] Note that the full control law includes both the locally lineargain term as well as the parameter rate of change term of equation (12).

[0070] Extending to Allow Design of Failure Detection Filter

[0071] U.S. Pat. No. ′119 shows the application to design a failuredetection filter (FDF) for the single input case. The previous sectionsshowed the general multi-input solution of the feedback (“FBK”)LTI'zation problem resulting in a set of LTI equations of motion inz-space, with the same number of inputs as the original co-ordinatesmodel. This section shows the application of the ideas described in the′119 patent to the multi input case, specifically, an example is givenfor the two input case. As will be appreciated by those skilled in theart, the more-than-two input case is a simple extension of the same formof equations.

[0072] With knowledge of the diffeomorsphism coefficients, it is nowpossible to define the FDF in transformed coordinates, which will be thesingle fixed-point design valid for the entire operating envelope of thesystem (e.g., aircraft). The failure detection filter is initiallydesigned at a nominal operating point in the flight envelope, using themodel described in physical coordinates, and taking advantage of theinsight gained by working in these coordinates. This design is thentransformed into the z-space coordinates to determine the transformedspace failure detection filter which is then unchanged for all operatingpoints in the flight envelope.

[0073] The z-space model is now, from equation (10), given by$\begin{matrix}{\overset{.}{z} = {{\begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}z} + {\begin{bmatrix}00 \\00 \\10 \\01\end{bmatrix}v}}} & (14)\end{matrix}$

[0074] and the FDF appropriately designed for the relevant failure modesyields the gain matrix H_(z) with the implemented system of thefollowing form:

{circumflex over ({dot over (z)})}=[A _(z) −H _(z) ]{circumflex over(z)}+H _(z) Φx _(measured) +B _(z)ν  (15)

[0075] or in physical vehicle coordinates,

{circumflex over ({dot over (x)})}=Φ ⁻¹ [A _(z) −H _(z) ]Φ{circumflexover (x)}+Φ ⁻¹ H _(z) Φx _(measured) +B _(x)ν  (16)

[0076] where the subscript z refers to the transformed state space, andthe subscript x refers to physical coordinates state space. Also,x_(measured) refers to the measured state variables. For example, themeasured values of roll rate (p), yaw rate (r), bank angle (φ) andsideslip (β). Since the full state vector is measured, each of the statevariables is available in physical coordinates.

[0077] Returning to FIG. 2, the multi-input feedback LTI'zation controllaw design example is given for the two input (aileron and rudderactuators) and two parameters (air density and dynamic pressure) case ofcontrolling the lateral dynamics of an aircraft, as shown in FIG. 1.Specifically, FIG. 2 show an example using reference bank angle (φref)and sideslip (βref) signals to control the aircraft 1. As will beappreciated by those of ordinary skill in the art, if a pilot wants theaircraft to fly right wing down ten degrees, he simply commands the bankangle of 10 degrees and the control law will cause the aircraft to flywith the right wing down 10 degrees. Likewise, if the pilot wants thenose of the aircraft to point 5 degrees to the left of the incomingairflow, this is achieved by commanding 5 degrees of sideslip, and thecontrol laws will cause the ailerons and rudder to move in such afashion as to deliver flight with the nose at 5 degrees to the incomingairflow.

[0078] As mentioned above, flight control laws are typically a pluralityof equations used to control flight in a predictable way. Flight controllaws are well known to those of ordinary skill in flight and vehiclecontrols and will not be described in greater detail herein. However,reference may be had to the text “AIRCRAFT DYNAMICS AND AUTOMATICCONTROL,” by McRuer, et al., Princeton University Press, 1973,incorporated herein by reference.

[0079] Known flight control laws for operating the aircraft rudder andaileron may be simplified as:

[0080] Rudder Control Law (Equation 17a):δ  r = −G_(RdrBeta)(Beta − Beta  Ref) − G_(RdrRollRate)(RollRate) − G_(RdrYawRate)(YawRate) − G_(RdrRoll)(Roll − RollREf) − G_(RdrBetaIntegrator)∫(Beta − BetaRef)t − G_(RdrRollIntegrator)∫(Roll − RollRef)t

[0081] where δr represents commanded rudder deflection angle, “G” termsrepresent rudder control law gains, Beta represents measured sideslip,RollRate represents measured roll rate, Roll represents measured bankangle, RollRef represents a reference bank angle and YawRate representsthe measured yaw rate.

[0082] Aileron Control Law (Equation 17b):δ  a = −G_(AilBeta)(Beta − Beta  Ref) − G_(AilRollRate)(RollRate) − G_(AilYawRate)(YawRate) − G_(AilRoll)(Roll − RollREf) − G_(AilBetaIntegrator)∫(Beta − BetaRef)t − G_(AilRollIntegrator)∫(Roll − RollRef)t

[0083] where δa represents commanded aileron deflection angle, “G” termsrepresent aileron control law gains, Beta represents measured side slip(e.g., measured with a sensor), RollRate represents measured roll rate,Roll represents measured bank angle, RollRef represents a reference bankangle and YawRate represents the measured yaw rate.

[0084] The integrator terms shape the closed loop vehicle dynamics bycompensating for the steady state error which typically results withoutthese extra terms.

[0085] Including the parameter rate of change term, yields the finalcontrol law: $\begin{matrix}{\begin{bmatrix}{\delta \quad a} \\{\delta \quad r}\end{bmatrix} = {\begin{bmatrix}{\delta \quad a} \\{\delta \quad r}\end{bmatrix}_{{\overset{.}{p}}_{i} = 0} - {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}{\sum\limits_{i = 1}^{q}\quad {\frac{\partial\Phi_{x}}{\partial p_{i}}{\overset{.}{p}}_{i}}}}}} & (18)\end{matrix}$

[0086] As discussed above, the “G” (gain) terms can be i) evaluatedoff-line, and stored in RAM look-up tables, and/or ii) evaluated inreal-time, as discussed above with respect to Eq. 13.

[0087] As seen in FIG. 2, reference bank angle (φref) and referenceside-slip (βref) signals 211 are compared (205) with sensor signalsreflecting the aircraft's current bank angle (φ) and side slip (β)respectively. These values are input (or utilized by) into equations 17,along with sensor signals representing the aircraft's actual roll-rate(p) and yaw-rate (r). The current dynamic air pressure (Q) and airdensity (ρ) are evaluated from sensor signals 201 and the correspondinggain values (e.g., “G_(i)”), implemented in one embodiment as RAMlook-up tables as functions of aircraft dynamic pressure and air density202, are applied to the control laws 206. The appropriate gain value isdetermined by interpolation between neighboring points in the lookuptables. The required number of gain look-up tables corresponds to thenumber of state variables plus any required integrals, multiplied by thenumber of control inputs (e.g., actuators). For example, the lateralaxis of an aircraft has four (4) state variables (i.e., sideslip, bankangle, roll rate, and yaw rate) and two integrals (sideslip error andbank angle error) for each actuator (i.e., rudder and aileron), for atotal of twelve (12) gain tables. Hence, in the lateral axis case, thereare twelve (12) corresponding look-up tables. If, for example, thelongitudinal axis were also considered, using an integral airspeed holdcontrol mode and actuating via the elevator, five more look-up tableswould be required. In the longitudinal axis case, there are four statevariables (i.e., angle of attack, pitch rate, pitch attitude, and trueairspeed) and one (1) integrator (i.e., the integral of airspeed minusairspeed reference). In another embodiment, the gain values can becalculated in real time, as discussed above with respect to Eq. 13.

[0088] The control gains (G_(i)) can be numerically evaluated in thecontrol law design process in z-space, using Linear Quadratic Regulators(LQR) theory, a well defined and widely known LTI control designtechnique. Pole placement or any other known LTI technique could also beused. As will be appreciated by those of ordinary skill in the art, LQRtheory provides a means of designing optimal control solutions for LTIsystems. A quadratic cost function, which penalizes state variableexcursions and actuator deflections in a weighted fashion, is solvedfor. The steady state solution yields a set of gains and a specific fullstate feedback control law which defines how aircraft motion is fed backto deflecting the control surfaces in order to maintain desired controlat a constant operating condition, i.e., constant parameter values.

[0089] At any specific operating condition, defined by ρ and Q, thecontrol gains are then transformed into x-space via equation (13),ignoring the dp/dt term for purposes of determining the gains, andstored in the RAM look-up tables. The dp/dt term is included as perequation (18), with relevant terms evaluated numerically, as discussedbelow. Alternatively, the values for the control gains can be determinedin real time, every computational cycle (e.g., every 60 milliseconds orfaster, depending on the required processing speed for the specificaircraft application), by two (2) dimensional interpolation for thecurrent air density and dynamic pressure that the aircraft isexperiencing at any point in time, as discussed above with respect toEq. 13. A second alternative is to fully determine the control commmandsin z-space, and then transform these to physical control commands usingequation (12) and solving for u.

[0090] Parameter rate of change.terms are evaluated numerically 203 foruse in equation 18 (204). Parameter rate of change terms compensate foror capture the varying rates of change of the operating conditions, asexperienced by the aircraft. For example, as an aircraft dives, the airdensity changes while the aircraft changes altitude. In order toaccommodate the effect of the changing density on the dynamic behaviorof the aircraft, the control system preferably accounts for the varyingair density.

[0091] An aircraft flying at high altitude and high true airspeed,typically exhibits much poorer damping of it's natural dynamics, than atlow altitude and low airspeed. This effect varies as the speed andaltitude varies, and the rate of change of the altitude and airspeedalso influence the dynamic behavior. In order to deliver similar closedloop behavior even whilst, e.g., decelerating, the rate of change ofdynamic pressure must be accounted for in the dp/dt terms of the controllaw.

[0092] Typically, parameters can be measured by means of a sensor, forexample, dynamic pressure can be directly measured, but the rate ofchange of the parameter is not typically directly measured. In thiscase, the rate of change value is determined numerically through one ofmany known methods of taking discrete derivatives, as will beappreciated by those skilled in the art. One example of such a numericalderivative evaluation is by evaluating the difference between a currentmeasurement and the previous measurement of a parameter, and dividing bythe time interval between the measurements. This quotient will give anumerically evaluated estimate of the rate of change of the parameter.These rate of change parameter values are evaluated in real time. Theother component of the parameter rate of change term, i.e., dφ/dp_(i) isalso numerically evaluated and can, however, be stored off-line inlook-up tables or evaluated in real time.

[0093] Blocks 204 and 206 are combined (in summing junction 212) toyield complete flight control commands for the rudder 103 and ailerons101 (in block 207). Signals are sent to the rudder and ailerons in 208,effecting control of the air vehicle dynamics (as shown in block 209).As mentioned above, sensors (210) feed back current roll rate, yaw rate,bank angle and slide-slip signals, and the above process is repeated,until the current measured bank angle and sideslip signals match therespective reference signals.

[0094]FIG. 3 is a flowchart depicting the software control carried outby the flight control computer 104. By way of illustration, a“decrabbing” maneuver is described in relation to FIG. 3. A decrabbingmaneuver is executed when an aircraft experiences a crosswind whilelanding. To perform the decrabbing maneuver, an aircraft on finalapproach to landing faces into the crosswind, and then at a moment priorto landing, adjusts so that the nose of the aircraft is pointed down(i.e., is parallel to and along) the runway. For this decrabbingmaneuver example, a 10 degree crosswind is imagined. To compensate forthe crosswind, the control system determines that in order to maintain asteady course (zero turn rate) whilst in a 10 degree sideslip condition,a 3 degree bank angle adjustment is also needed. Hence, in this example,the reference sideslip (βref) and bank angle (φref) are 10 degrees and 3degrees, respectively.

[0095] Referring to FIG. 3, in step S1, the reference sideslip (βref)and bank angle (φref) are input as reference signals. In step S2, thecurrent air density (ρ) and dynamic pressure (Q) are input from sensors.In step S3, the rudder (δr) and aileron (δa) control laws are read frommemory. In step S4, the roll rate (p), yaw rate (r), bank angle (φ) andsideslip (β) signals are input from sensors.

[0096] In step S5, βref and φref are compared with β and φ signals fromsensors. In step S6, signals p and r, the comparison from step S5,sideslip and bank angle integrators, and control gains (Gi) are appliedto the rudder (δr) and aileron (δa) control laws. These control gains(G_(i)) are preferably off-line resolved to capture z-spacetransformations. In step S7, parameter rates of change terms aregenerated. Note that if the vehicle parameter rates of change are verysmall, these terms would be very small in magnitude and as such may beeliminated from the control law.

[0097] In step S8, the parameter rates of change generated in step S7are applied to the rudder (δr) and aileron (δa) control laws. In stepS9, a control signal is output to the aileron 101 and rudder 103actuators, effecting control of the aircraft. The control lawintegrators are incremented in step S10. In the de-crabbing maneuverexample, the logic flow continues adjusting the rudder and aileron untilβ and φ approximate 10 and 3 degrees, respectively. In this manner, thecontrol system compensates for the 10-degree crosswind, by turning theaircraft's nose parallel to the runway just prior to landing. Ahigh-level control function may be implemented with the above-describedcontrol system to effect landing. For example, the high-level controlcould be a pilot or automatic control algorithm such as an autoland.

[0098]FIG. 4 is a block diagram showing the relationship between thevarious sensors, actuators and the flight control computer 104. As canbe seen, flight control computer 104receives input from various sensors,including airspeed 105 a, altimeter 105 b, yaw rate 105 c, bank angle105 d, side slip 105 e, angle of attack 105 f, pitch rate 105 g, pitchattitude 105 h, roll rate 105 i and Nth sensor 105 n. The various sensorsignals are inserted into the appropriated flight control laws and theoutputs are actuator command signals, such as to the throttle 106 a,elevator 106 b, aileron 106 c, rudder 106 d, and Mth actuator 106 m.

[0099] Applying Feedback LTI'zation to Designing Control Laws

[0100] An example of the control law design techniques according to thepresent invention will now be described with respect to FIGS. 5a-30.Essentially, the design process involves transforming the coordinatesfor the vehicle equations of motion into z-space. This step has beendetailed in equations 1-18, above. Known LTI control design techniquesare used as a framework for the control gains design process. Parametervalues at a few desired design points in the operating envelope areselected. LTI design techniques are applied to the physical LTI modelsat the (few) selected parameter values, to yield desired closed loopdynamics. These designs are transformed into the transformed coordinates(in z-space) to yield z-space gains that give desired closed loopbehavior for the controlled system. If more than one design point wasselected, these z-space gains are linearly interpolated over theoperating envelope, otherwise the gains are constant in z-space for thefull envelope. Finally, a discrete number of parameter values,corresponding to lookup table axes, is selected, and the reversetransformation applied to define the physical coordinates lookup gaintables for use as discussed above with respect to FIGS. 2 and 3, forexample.

[0101] By way of example, FIGS. 7-18 show lateral auto pilot gains in3-D plot form for the lateral axis of the Perseus 004 aircraft aircraftover a flight envelope of sea level to 22 km altitude and 20 m/s to46.95 m/s IAS). Each figure illustrates an auto pilot gain with respectto air density (kg/m³) and dynamic pressure (Pa). In particular, FIG. 7illustrates side slip to aileron feedback gain; FIG. 8 illustrates rollrate to aileron feedback gain; FIG. 9 illustrates yaw rate to aileronfeedback gain; FIG. 10 illustrates roll attitude to aileron feedbackgain; FIG. 11 illustrates side slip integrator to aileron feedback gain;FIG. 12 illustrates roll integrator to aileron feedback gain; FIG. 13illustrates side slip to rudder feedback gain; FIG. 14 illustrates rollrate to rudder feedback gain; FIG. 15 illustrates yaw rate to rudderfeedback gain; FIG. 16 illustrates roll attitude to rudder feedbackgain; FIG. 17 illustrates side slip integrator to rudder feedback gain;and FIG. 18 illustrates roll attitude integrator to rudder feedbackgain. Each of the represented gains is illustrated in physicalcoordinates.

[0102] FIGS. 19-30 are corresponding matrix numerical gain lookup tablesfor the lateral axis of the Perseus 004 aircraft over a flight envelopeof sea level to 22 km altitude and 20 m/s to 46.95 m/s IAS. FIGS. 19-30provide the numerical data for FIGS. 7-18, respectively. The format foreach of FIGS. 19-30 is as follows: the first row is a dynamic pressurelookup parameter in Pa, the second row is a density lookup parameter inkg/m{circumflex over ( )}3, and the remainder of rows are gain values.

[0103] Together, FIGS. 7-30 illustrate part of the design process fordetermining gains. In this example, optimal LQR designs are generated atfour discrete design points corresponding to the four corners of thecontrol design flight envelope (i.e., sea level to 22 km altitude and 20m/s to 46.95 m/s IAS). The four design points are thus low speed at highdensity; high speed at low density; low speed at low density; and highspeed at high density. Feedback LTI'zation is used to map these fourdesigns into smooth gain scheduling look-up tables at 110 points in adensity—dynamic pressure space. Each of the four (4) designs is done ata selected steady state flight condition (density, speed combination),in physical co-ordinates, since the designer readily understands these.The resulting gains (physical) at each design point are then transformedinto z-space, and by reversing this transformation (i.e., from z tophysical co-ordinates) at any parameter values, the physical coordinateslookup table values are populated. Typically, a selected matrix ofparameter values is defined, and the lookup tables are populated forthese parameter values. The process of populating the gain tables simplyexecutes equation 11, and does not require the designer to intervene ateach table lookup parameter value. This is a major reason for thesavings in design effort and time, namely that only a few design pointsare required, and then the full envelope is covered by appropriatetransformation using equation 11. Note that this results in physicalgains that can be used in real time in the control law. It is alsofeasible to perform real time reverse co-ordinates transformations fromthe z-space gains, at every time step, to determine the x-space gains inreal time. In this case, the lookup tables store z-space gains and notphysical x-space gains.

[0104] Root loci and step response data are used to evaluate performanceand robustness during the design process. In particular, FIGS. 5a-5 dshow the full envelope design results for a lateral auto pilot controlsystem design for the Perseus 004 aircraft. In particular, overlaiddiscrete bode plots (i.e., FIGS. 5c and 5 d) and step responses (i.e.,FIGS. 5a and 5 b) are shown for all combinations of air density anddynamic pressure in the gain tables covering the design envelope of sealevel to 22 km above sea level and 20 m/s to 46.95 m/s IAS. Thisillustrates how the design can be used to achieve similar and wellbehaved closed loop performance across the full envelope of operation,while requiring only a very small number of design points—namely fourdesign points in this example.

[0105]FIG. 6 illustrates the S-plane root loci for closed loop and openloop lateral dynamics over the entire flight envelope for the Perseus004 aircraft, at the discrete density and dynamic pressure values in thelookup table. Open loop poles are circles, and closed loop poles arecrosses. Closed loop poles all lie inside the 45 degree sector from theorigin about the negative real axis, which is the design criterion forgood damping characteristics. By design, the closed loop modal frequencymagnitudes are not increased significantly over the open loop values.This reduces the danger of actuator saturation in normal envelopeoperation, as well as danger of delays due to too high closed loop modefrequencies for the sample period of 60 ms. The design goal of betterthan 70 percent damping of all modes is achieved, without significantlyaltering the modal frequencies.

[0106] This design example is particularly tailored between only fourdesign points, namely one at each corner of the density/dynamic pressurespace, which defines the flight control design envelope. At each of thefour design points, the well known LQR control design algorithm is usedfor determining the controller gains, as discussed above. The four pointdesigns are then transformed into a new set of coordinates, i.e., the socalled z-space via feedback LTI'zation routines, which are then furtherinvoked to determine the physical gains over the entire flight envelope,based on linear blending in z-space of the four point designs. The fourpoint designs yield four sets of gains in z-space, and these are simplylinearly interpolated between the four design points to provide linearlyblended gains in z-space. This technique achieves an approximatelylinear variation of closed loop bandwidth over the design envelope.

[0107] By transforming the single point design gains into z-space, wherethe definition of z-space forces the system to be LTI, the control gainsat any other operating condition can be easily determined by simplyreversing the co-ordinates transformation process. This allows thesingle design point to be transformed into z-space and reversetransformed to an infinite number of operating conditions different fromthe design point. Including the parameter rate of change terms then alsoallows transition between design points without disturbing the closedloop behavior, in the sense that the closed loop characteristics remainconstant. It is this coordinates transformation step that allows a verysmall number of design points to cover the full operating envelope ofthe vehicle. Those of ordinary skill in the art will appreciate that the“few” design point case has very similar attributes to the single designpoint case.

[0108] Thus, what has been described is a control system (and method) tocontrol a dynamic device or system with multiple control inputs andmultiple parameter dependencies. An efficient method of control lawdesign has also been described for such multi-input, parameter dependentdynamic systems.

[0109] The individual components shown in outline or designated byblocks in the drawings are all well known in the arts, and theirspecific construction and operation are not critical to the operation orbest mode for carrying out the invention.

[0110] While the present invention has been described with respect towhat is presently considered to be the preferred embodiments, it is tobe understood that the invention is not limited to the disclosedembodiments. To the contrary, the invention is intended to cover variousmodifications and equivalent arrangements included within the spirit andscope of the appended claims. The scope of the following claims is to beaccorded the broadest interpretation so as to encompass all suchmodifications and equivalent structures and functions.

I claim the following:
 1. An automatic control system for controlling adynamic device having device characteristics, the control systemincluding (i) a plurality of sensors and (ii) a plurality of controllaws stored in a memory, said system comprising: receiving means forreceiving a plurality of status signals and a plurality of currentexternal conditions signals from the sensors, and for receiving aplurality of reference signals; and processing structure for: (i)selecting and applying gain schedules to update the control laws,wherein the gain schedules correspond to the received current externalconditions signals, and the gain schedules are generated by transformingthe device characteristics into a multi-input linear time invariantcoordinates system; (ii) determining control law parameter rates ofchange and applying the parameter rates of change to update the controllaws; (iii) applying the received status signals from the device toupdate the control laws; and (iv) controlling the device based on theupdated control laws.
 2. A method of designing flight control laws usingmulti-input, multi-output feedback LTI'zation, said method comprisingthe following steps: determining coordinates for flight vehicleequations of motion; transforming the coordinates for the flight vehicleequations of motion into a multi-input linear time invariant system;establishing control laws yielding the transformed equations of motionLTI; adjusting the control laws to obtain a desired closed loop behaviorfor the controlled system; and converting the transformed coordinatescontrol laws to physical coordinates.
 3. A method of controlling adynamic device having device characteristics, the device including aplurality of sensors for receiving device status signals and parametersignals and a plurality of control laws stored in a memory, the deviceoperating in an environment with a plurality of external operatingconditions, said method comprising the steps of: transforming the devicecharacteristics into a multi-input linear time invariant system;selecting and applying physical gain schedules to the control laws, thegain schedules corresponding to current external operating conditionssignals; determining and applying parameter rates of change to updatethe control laws; applying the status signals to update the controllaws; converting the transformed coordinates control laws to physicalcoordinates; and controlling the device based on the updated controllaws.
 4. Computer executable software stored on a computer or processorreadable medium, the code for developing control laws for a dynamicdevice having device characteristics, the code comprising; code totransform the device characteristics into a multi-input linear timeinvariant system; code to establish control criteria yielding the devicecharacteristics in the transformed coordinates LTI; code to define atleast one design point in the multi-input linear time invariant system;code to adjust the transformations to corresponding with the designpoint(s); code to develop a physical coordinates control lawcorresponding to the adjusted transformations; and code to apply reversetransformations to cover the full design envelope.
 5. A multi-inputparameter dependent control system for controlling an aircraft, saidsystem comprising: receiving means for receiving a plurality of aircraftstatus signals and for receiving a plurality of current parameterssignals; memory having at least one region for storing computerexecutable code; and a processor for executing the program code, whereinthe program code includes code responsive to: (i) transform the aircraftcharacteristics into a multi-input linear time invariant system; (ii)select and apply gain schedules to flight control laws in transformedcoordinates, the gain schedules corresponding to the received currentparameter signals; (iii) determine parameter rates of change, and toapply the parameter rates of change to the flight control laws; (iv)apply the received aircraft status signals to the flight control laws;(v) convert the transformed coordinates control laws to physicalcoordinates; and (vi) control the aircraft based on the updated flightcontrol laws.
 6. A method of determining the operational status of amulti-input, multi-parameter dependent control system for controlling adynamic device, said method comprising the steps of: receiving aplurality of current status signals representing current statuses of thedevice and a plurality of control signals for effecting control of thedevice; transforming the current status signals and the control signalsto a multi-input linear time invariant system; estimating an expectedbehavior of the device using the transformed current status signals andthe transformed control signals; and determining the operational statusof the control system by comparing the expected behavior of the deviceto the actual behavior of the device.
 7. A multi-parameter dependentcontrol system for controlling a dynamic device having multiple inputs,said system comprising: receiving means for receiving a plurality ofcurrent state signals representing current states of the device and aplurality of control signals for effecting control of the device;transforming means for transforming the current state signals and thecontrol signals to a linear time invariant system; estimating means forestimating an expected behavior of the device using the transformedcurrent state signals and the transformed control signals and forgenerating estimate signals corresponding to the expected behavior; anddetermining means for determining the operational status of the controlsystem by comparing the expected behavior of the device to the actualbehavior of the device.
 8. Apparatus for detecting a failure in a flightcontrol device having multiple inputs and operating in an environmenthaving multiple parameters, comprising: processing structure for (i)receiving time-varying status signals from the flight control device,(ii) providing reference signals corresponding to the flight controldevice, (iii) transforming both the status signals and the referencesignals to a linear time invariant coordinate system, (iv) calculatingflight control device estimated signals based on the transformed statussignals and the transformed reference signals, (v) transforming thecalculated estimate signal in a physical coordinate system, and (vi)detecting an error in the flight control device when a differencebetween the transformed estimate signals and the status signals exceed apredetermined threshold.
 9. A method of designing control laws for adynamic device, the device accepting multiple inputs and generatingmultiple outputs and operating in an environment having varyingparameters, characteristics of the device being definable by equationsof motion, said method comprising the steps of: transforming coordinatesfor the equations of motion into a linear-time invariant system; andestablishing a closed loop behavior in the linear time invariant system,wherein said establishing step generates gain schedules for controllingthe equations of motion throughout an operational envelope of thedevice.
 10. A method of designing control laws for a dynamic device, thedevice accepting multiple inputs and generating multiple outputs,characteristics of the device being definable by equations of motion,said method comprising the steps of: defining an operating envelopeincluding varying parameters throughout the envelope; determiningcontrol designs for a plurality of discrete points in the envelope;transforming the plurality of discrete point designs to a linear timeinvariant system (z-space) to provide corresponding gains in z-space andinterpolating between the gains in z-space to provide linearly blendedgains; and inversely transforming the z-space linearly blended gains tophysical space.
 11. The method according to claim 10, wherein thephysical space gains respectively correspond to any operating conditionsthroughout the operating envelope, not necessarily the designconditions.
 12. A method of designing control laws for a dynamic device,the device accepting multiple inputs and generating multiple outputs,characteristics of the device being defined by at least one control law,said method comprising the steps of: determining in physical space gainscorresponding to a discrete operating condition, the gains for use inthe control law; transforming the gains obtained at the discreteoperating condition to a linear time invariant system (z-space) andobtaining corresponding gains in z-space; and inversely transforming thez-space gains into physical space to correspond to a plurality ofoperating conditions throughout an envelope in which the deviceoperates.
 13. A method of designing control laws for a dynamic device,the device accepting multiple inputs and generating multiple outputs,the device operating in an environment having multiple parameters, saidmethod comprising the steps of: defining a control law comprisingderivative, proportional and integral gains and device inputs; anddeveloping gain schedules to be applied to the control law so as toaccommodate varying parameters, wherein transforming devicecharacteristics into a liner-time invariant system generates the gainschedules and provides a stable closed loop behavior for the transformedcharacteristics.
 14. A method of controlling a device havingmulti-inputs and operating in an environment having multiple operatingparameters, said method comprising: defining a control law to describecharacteristics of the device, the control law comprising derivative,proportional and integral gains; evaluating in real-time z-space gainsfor the control law according to current operating parameters; andtransforming the z-space gains to physical space for use in the controllaw.
 15. A method of controlling a dynamic device having multi-inputsand operating in an environment having multiple operating parameterscomprising: providing a feedback LTI'zation control algorithm in theform of Eq. 11, in which arbitrary rates of change of the parameters areaccommodated; determining in a linear time-invariant coordinates systemgains corresponding to the control algorithm for discrete operatingconditions; transforming the gains into physical space; and applying thegains to control the dynamic device.
 16. A method of providing controlof a dynamic device, comprising: defining a control law to control thebehavior of the dynamic device, the control law comprising at least onegain variable; transforming the gain variable to a linear time invariantsystem (z-space) and determining a corresponding z-space gain; mappingthe z-space gain into a plurality of physical space gains that eachcorrespond respectively to a plurality of operating conditions; storingin memory the physical space gains; and accessing the stored gains inreal-time for use in the control law under particular operatingconditions.
 17. A method of applying LTI design techniques to atransformed coordinates mathematical model of a dynamic device to yielda desired closed loop behavior for the model, the control law includingfeedback gains, said method comprising the steps of: designing thefeedback gains in physical coordinates at a selected operating conditionof the dynamic device; using the transformed coordinates to map thefeedback gains into z-space; and reverse mapping the gains via acoordinates transformations and feedback LTI'ing control laws todetermine physical coordinates control laws for operating conditionsother than at design conditions.